Problem: Gustavo and Yuki were asked to find an explicit formula for the sequence $19,0,-19,-38,...$, where the first term should be $h(1)$. Gustavo said the formula is $h(n)=19-19n$. Yuki said the formula is $h(n)=-19+19n$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Gustavo (Choice B) B Only Yuki (Choice C) C Both Gustavo and Yuki (Choice D) D Neither Gustavo nor Yuki
Solution: The general explicit formula for arithmetic sequences is ${a_1}+{d}(n-1)$, where ${a_1}$ is the first term and $ d$ is the common difference. The first term is ${19}$ and the common difference is ${-19}$. ${-19\,\curvearrowright}$ ${-19\,\curvearrowright}$ ${-19\,\curvearrowright}$ ${19},$ $0,$ $-19,$ $-38,...$ We get the following formula. $h(n)={19}{-19}(n-1)$ We can now see that $h(n)=19-19n$ is not a correct formula, because the constant difference is added one extra time for each term. For instance, according to this formula, the value of the first term would be: $h(1)=19-19\cdot1=0$. However, according to our table of values, $h(1)=19$. So Gustavo is definitely wrong. What about Yuki? We can see that $h(n)=-19+19n$ is also not a correct formula, because the constant difference according to this formula is $19$, while the actual constant difference is $-19$. So Yuki is also wrong. Neither Gustavo nor Yuki got a correct explicit formula.